In topological data analysis (TDA), one often studies the shape of data by constructing a
filtered topological space, whose structure is then examined using persistent homology …

This paper introduces persistent homology, which is a powerful tool to characterize the
shape of data using the mathematical concept of topology. We explain the fundamental idea …

The notion of generalized rank in the context of multiparameter persistence has become an
important ingredient for defining interesting homological structures such as generalized …

We define a class of invariants, which we call homological invariants, for persistence
modules over a finite poset. Informally, a homological invariant is one that respects some …

In this work, we propose a new invariant for 2D persistence modules called the compressed
multiplicity and show that it generalizes the notions of the dimension vector and the rank …

In topological data analysis, two-parameter persistence can be studied using the
representation theory of the 2d commutative grid, the tensor product of two Dynkin quivers of …

Comments• page viii, bottom of page: the following names should be added to the
acknowledgements:-Peter Landweber had an invaluable contribution to these notes. First …

This paper addresses two questions:(a) can we identify a sensible class of 2-parameter
persistence modules on which the rank invariant is complete?(b) can we determine …

Persistent homology is currently one of the more widely known tools from computational
topology and topological data analysis. We present in this note a brief survey on the …

A fundamental challenge in multiparameter persistent homology is the absence of a
complete and discrete invariant. To address this issue, we propose an enhanced framework …